Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Tournament prize winnings A bowling tournament is handicapped so that all 80 bowlers are equally matched. The tournament prizes are listed in the table. \(\begin{tabular}{|c|ccccc|} \hline Place & \)1 \mathrm{st}\( & 2 nd & 3rd & 4th & 5 th-10th \\ \hline Prize & \)\$ 1000\( & \)\$ 500\( & \)\$ 300\( & \)\$ 200\( & \)\$ 100\( \\ \hline\) \end{tabular} Find the expected winnings for one contestant.

Step-by-Step Solution

Answer

The expected winnings for one contestant is $32.5.

Step 1: Understand the Distribution of Prizes

There are 80 bowlers in total and the prizes are allocated as follows: 1st place receives $1000, 2nd place receives $500, 3rd place gets $300, 4th place receives $200, and 5th to 10th places each receive $100.

Step 2: Calculate the Probability of Winning Each Prize

Since all bowlers are equally matched, each bowler has an equal chance of winning any prize. Therefore, the probabilities for each position are: \( \frac{1}{80} \) for each of the top 4 places, and \( \frac{6}{80} \) for placing 5th to 10th (since there are 6 possible places for this prize).

Step 3: Calculate the Expected Value

The expected value is calculated by multiplying the prize amount by its probability and summing these products. Therefore,\[\text{Expected Winnings} = 1000 \times \frac{1}{80} + 500 \times \frac{1}{80} + 300 \times \frac{1}{80} + 200 \times \frac{1}{80} + 100 \times \frac{6}{80}.\]

Step 4: Compute the Total Expected Winnings

Compute each term:- \(1000 \times \frac{1}{80} = 12.5\)- \(500 \times \frac{1}{80} = 6.25\)- \(300 \times \frac{1}{80} = 3.75\)- \(200 \times \frac{1}{80} = 2.5\)- \(100 \times \frac{6}{80} = 7.5\)Add all these values together:\[12.5 + 6.25 + 3.75 + 2.5 + 7.5 = 32.5.\]

Step 5: Conclusion

Thus, the expected winnings for one contestant in this tournament is $32.5.

Key Concepts

Probability

Understanding probability is key to solving exercises involving potential winnings, like in a bowling tournament. Probability helps us determine the likelihood of each outcome.
In a tournament with 80 equally matched bowlers, each one has the same chance to win. This implies that the probability of a bowler securing any specific place is based on the number of places available.
- For the 1st to 4th positions, only one spot is available for each place. Therefore, the probability of winning one of these prizes is \( \frac{1}{80} \). - For the 5th to 10th positions, six places share the same prize. Hence, the probability of ending in this range is \( \frac{6}{80} \).
By understanding these probabilities, we can further calculate expected winnings in the tournament.

Prizes

Prizes in a bowling tournament serve as an excellent motivation for participants. Each position in the tournament is associated with a specific prize. In our scenario:
- The 1st prize is $1000. - The 2nd prize is $500. - The 3rd prize is $300. - The 4th prize is $200. - The 5th to 10th prizes are $100 each.
These prizes reflect how the tournament rewards performance. Calculating the expected winnings requires understanding how each of these prizes impacts the overall average—a calculation directly influenced by their probability of being won.

Bowling Tournament

A bowling tournament is an exciting competitive event where individual skill plays a crucial role. In our scenario, it's set up to be handicapped, meaning all 80 bowlers have been adjusted to be equally matched.
This system aims to level the playing field so that everyone has an equal shot at winning the various prizes available.
Thus, each contestant competes under the same conditions, and any bowler could end up in the top places just as easily as another, based solely on their performance that day.

Expected Winnings

Calculating the expected winnings of a contestant in a bowling tournament involves combining probability with prize distribution.
The expected value is a weighted average of all possible winnings, taking into account the likelihood of each prize being won. Here's how we compute it:
1. Each possible prize is multiplied by its respective probability: - Winning 1st place: \(1000 \times \frac{1}{80} = 12.5\) - Winning 2nd place: \(500 \times \frac{1}{80} = 6.25\) - Winning 3rd place: \(300 \times \frac{1}{80} = 3.75\) - Winning 4th place: \(200 \times \frac{1}{80} = 2.5\) - Winning a 5th-10th place: \(100 \times \frac{6}{80} = 7.5\)2. Summing these values gives the expected winnings: \[12.5 + 6.25 + 3.75 + 2.5 + 7.5 = 32.5\]Thus, each contestant can expect an average of $32.5 in winnings by competing in this tournament. This simplified view helps participants understand the tournament's payout structure and manage their expectations.